Solution: Solve the system. From the second equation, $ x = y + 2 $. Substitute into the first: $ 2(y + 2) + y = 10 \Rightarrow 2y + 4 + y = 10 \Rightarrow 3y = 6 \Rightarrow y = 2 $. Then $ x = 2 + 2 = 4 $. The intersection is $ \boxed{(4, 2)} $.

February 22, 2026

$Solution: Solve the system. From the second equation, $ x = y + 2 $. Substitute into the first: $ 2(y + 2) + y = 10 \Rightarrow 2y + 4 + y = 10 \Rightarrow 3y = 6 \Rightarrow y = 2 $. Then $ x = 2 + 2 = 4 $. The intersection is $ \boxed{(4, 2)} $.$

Solve the System of Equations Step-by-Step: A Clear Guide to Finding the Solution

Solving a system of equations is a fundamental skill in algebra, essential for math students, engineers, and anyone working with real-world modeling. Understanding how to find the point where two equations intersect—also known as the solution to the system—is key to mastering linear relationships. In this article, we break down a classic linear system with clear steps to solve for $ x $ and $ y $.

Understanding the Problem

Suppose you’re given two equations:

$ x = y + 2 $
$ 2x + y = 10 $

These represent two lines on a coordinate plane. The goal is to find the single point $ (x, y) $ where both equations are true simultaneously—this is the solution to the system.

Step-by-Step Solution

Step 1: Substitute Using the Second Equation

We start with the second equation: $$ 2x + y = 10 $$

We’re given a direct relationship between $ x $ and $ y $ in the first equation: $$ x = y + 2 $$

Instead of solving for one variable at a time, substitution simplifies the process. Replace $ x $ in the second equation with $ y + 2 $: $$ 2(y + 2) + y = 10 $$

Step 2: Expand and Simplify

Now expand the expression: $$ 2y + 4 + y = 10 $$

Combine like terms: $$ 3y + 4 = 10 $$

Subtract 4 from both sides: $$ 3y = 6 $$

Divide by 3: $$ y = 2 $$

Step 3: Find $ x $ Using the First Equation

Now that we know $ y = 2 $, substitute back into the original equation $ x = y + 2 $: $$ x = 2 + 2 = 4 $$

Step 4: Interpret the Solution

The solution to the system is the ordered pair where both equations meet: $$ oxed{(4, 2)} $$

This means when $ x = 4 $ and $ y = 2 $, both equations are satisfied.

Why This Method Works

Substitution leverages the relationship between variables to reduce a system from two equations to one. This makes solving much simpler—especially helpful with harder systems involving more variables or nonlinear equations. It’s a foundational technique used across science and engineering fields.

Final Thoughts

Solving systems step-by-step removes confusion and builds confidence. The example above shows how a simple substitution leads directly to the answer. Whether you’re a student learning basic algebra or a professional working through complex models, mastering this method empowers you to tackle increasingly challenging problems with clarity and precision.

Remember: Always check your solution by plugging $ x $ and $ y $ back into both original equations—verification ensures accuracy!

By understanding substitution and stepwise reduction, finding intersections becomes intuitive. Keep practicing, and soon solving systems will feel natural and quick.

--- Keywords: solving linear systems, substitution method, algebraic equations, step-by-step solution, intersection point (x,y), algebra tutorial